Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 63

Capitolo 5, Problema 63

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/2)(log x + log y)

Guida verificata passo dopo passo

Start with the given expression: \(\frac{1}{2}(\log x + \log y)\).

Use the logarithm property that the sum of logarithms is the logarithm of the product: \(\log x + \log y = \log(xy)\).

Rewrite the expression as \(\frac{1}{2} \log(xy)\).

Apply the power rule of logarithms, which states that \(a \log b = \log(b^a)\), to move the coefficient inside the logarithm: \(\frac{1}{2} \log(xy) = \log((xy)^{\frac{1}{2}})\).

Express the final condensed form as a single logarithm: \(\log \sqrt{xy}\).

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Properties of Logarithms

Properties of logarithms include rules such as the product rule (log a + log b = log(ab)), the power rule (k log a = log(a^k)), and the quotient rule (log a - log b = log(a/b)). These rules allow combining or breaking down logarithmic expressions to simplify or condense them.

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Change of Base Property

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying the properties of logarithms to combine sums, differences, and coefficients into one logarithmic term with coefficient 1, making the expression simpler and easier to evaluate.

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Expand & Condense Log Expressions

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing logarithms of numbers that are powers of the base, such as log base 10 of 100 = 2. Simplifying expressions using properties can help rewrite logarithms in terms of known values, enabling exact evaluation without computational tools.

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Evaluate Logarithms

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